Ellipsoidal Inner Approximations of Region of Attraction

• Ellipsoidal inner approximations are convex, computationally efficient subsets of a system's region of attraction found using quadratic Lyapunov functions.
• Despite potential conservatism, ellipsoidal approximations offer computational efficiency using LMI/SOS, proving practical for certifying stability in systems like power grids.
• Ellipsoidal methods are foundational to a hierarchy of polynomial approximations, with research focused on scalability, data integration, and advanced shape fitting.

Ellipsoidal inner approximations of the region of attraction (ROA) are a classical and enduring paradigm in nonlinear stability analysis, in which the set of states guaranteed to asymptotically approach a target set or equilibrium is described or contained by ellipsoidal (i.e., quadratic-form) subsets of the state space. These ellipsoidal regions arise naturally from Lyapunov theory, optimization techniques, and set-approximation methods, and serve as computationally efficient, convex inner bounds for both analysis and synthesis tasks in dynamical systems. The following sections synthesize principal methods, formulations, convergence properties, and applications as documented in the research literature.

1. Mathematical Foundations and Lyapunov Certificate

At the heart of ellipsoidal inner approximations lies the Lyapunov direct method. For a system

$\dot{x} = f(x),$

a quadratic function $V(x) = x^\top P x$ with $P \succ 0$ is a Lyapunov function if $V(x) > 0$ for $x \neq 0$ and $\dot{V}(x) < 0$ in a region containing the equilibrium. The sublevel set

$E = \{ x \in \mathbb{R}^n : x^\top P x < 1 \}$

is an ellipsoid, guaranteed invariant and contained in the ROA if $\dot{V}(x) < 0$ for all $x \in E\setminus\{0\}$. This certifies that trajectories from any initial condition in $E$ converge to the equilibrium.

This principle extends naturally to both unconstrained and constrained polynomial systems, forming the basis for convex relaxations and optimization-based set computations (1210.3184).

2. Convex Optimization Approaches: SOS and LMI Relaxations

Advances in semidefinite programming (SDP) and sum-of-squares (SOS) optimization enable systematic computation of ellipsoidal (and higher-degree polynomial) inner approximations for polynomial systems. The framework operates by:

1. Encoding invariance or contraction conditions (e.g., $\dot{V}(x) < 0$ in $E$) as SOS constraints, directly translatable to LMIs (1210.3184, 1208.1751).
2. Solving for the largest ellipsoid (parameterized by $P$) certifiably contained within the ROA, typically maximizing $\det P$ or $\mathrm{vol}(E)$ subject to invariance (2203.13071).
3. Extending to generalized polynomial sublevel sets by increasing the degree of $V$, yielding less conservative, possibly non-ellipsoidal approximations, but ellipsoidal cases arise when $V$ is quadratic (1210.3184, 2103.12825).

The entire process is encapsulated in convex SDP solvers (e.g., SeDuMi, MOSEK), allowing for numerically robust and scalable implementation for moderate system sizes.

3. Hierarchy, Convergence Properties, and Practical Limitations

The sum-of-squares approach is structured as a hierarchy in which the degree of the Lyapunov candidate (and, accordingly, the tightness and possibly the complexity of the region) increases (1208.1751, 1210.3184). For $\deg(V) = 2$, the inner approximations are ellipsoidal.

Key properties include:

• Convergence: As the hierarchy degree increases, the inner approximations (sublevel sets) collectively exhaust the true ROA in volume—i.e., the Lebesgue measure of points in the ROA not covered by the polynomial sublevel sets tends to zero (1210.3184, 2103.12825, 1903.04798).
• Conservativeness: While quadratic (ellipsoidal) approximations are guaranteed, they may be substantially conservative for systems with a non-convex or highly nonlinear ROA. Higher-degree polynomials mitigate this at the expense of computational burden (1210.3184, 2103.12825).
• Scalability: The number of LMI variables grows quickly with system dimension and polynomial degree, making the ellipsoidal (quadratic) case the practical workhorse in high-dimension, but more complex shapes become tractable only for low-to-moderate $n$.

4. Extensions and Generalizations

The ellipsoidal approximation methodology, while classic, is tightly interwoven with several major extensions:

• Star-convex Sets: For sets defined by intersecting polynomials ($\mathcal{X} = \{x : g_i(x) \leq 1\}$), convex programming can be used to find the largest ellipsoid contained in $\mathcal{X}$, or, in the more general setting, the maximum-volume inner polynomial sublevel set. For star-convex sets, scale-invariant objectives—such as minimizing the scale $s \geq 1$ in $$\mathcal{F} \subseteq \mathcal{X} \subseteq s\mathcal{F}$$—provide tighter inner approximations (2203.13071).
• Control Systems and Constraints: The framework supports controlled polynomial systems (with state and input constraints), provided that the invariance conditions can be encoded (e.g., by restricting controls in the occupation measure framework) (1310.2213).
• Stochastic and Uncertain Systems: Quadratic inner approximations generalize to the robust or stochastic setting via Lyapunov analysis in the appropriately lifted (e.g., polynomial chaos expanded) system (1911.00252), allowing ellipsoids to certify mean or moment ROA (but see the data for richer alternatives in those cases).
• Alternative Set Shapes: Shifting the ellipsoid center (i.e., using shifted quadratic forms), or combining multiple ellipsoids via union or convex hull, extends the flexibility—particularly for disconnected, skewed, or unbounded ROAs (2207.05421).

5. Numerical Examples and Benchmark Results

Numerical studies demonstrate the method’s applicability and practical tightness:

• Polynomial Benchmarks: For univariate and low-dimensional multivariate polynomial systems, the maximal ellipsoidal sublevel set can be computed, and its relative error with respect to numerically estimated true ROA tabulated (e.g., errors of a few percent for moderate degrees; higher accuracy as degree increases) (1210.3184).
• Transitional Flow Models: For reduced-order fluid flow models, ellipsoidal QC-based ROA estimation is computationally efficient and substantially less conservative than earlier sphere-based QC approaches or direct-adjoint looping (see Table below from (2103.05426)):

Method	Wall Time (WKH model, 4D)	Conservatism (versus best SOS)
SOS	117 s	1 $\times$
QC/ellipsoid	3 s (Algorithm A)	1.75 $\times$

• Power Systems: In large-scale power network models, explicit ellipsoidal or box-type ROA approximations enable tractable, safe stability certificates suitable for scalable real-time operation (2009.12345).

6. Advantages, Limitations, and Alternatives

Property	Ellipsoidal Inner Approximation	Alternatives/Extensions
Convexity	Yes	Generally yes for polynomial SOS
Certifiability	Lyapunov theory	Lyapunov/SOS for polynomials
Computational Cost	Low (quadratic), moderate (poly)	High for large degree/high $n$
Shape Flexibility	Convex, symmetric	More flexible (star-convex, unions)
Conservative vs. tight	Conservative for nonlinear ROAs	Tighter for higher-degree polynomials

A plausible implication is that while ellipsoidal inner approximations are computationally attractive and serve as a foundation for more complex invariant set approximations, they may be overly conservative for systems where the true ROA exhibits significant non-convexity, directional skew, or multiple disconnected components. In those cases, polynomial (SOS) and algorithmic extensions (R-function composition, shifted/unioned ellipsoids, data-driven methods) are pivotal.

7. Future Research and Practical Deployment

Recent and ongoing research has focused on:

• Enhancing scalability of LMI/SOS-based inner approximations through sparsity exploitation, alternative bases (Chebyshev, etc.), and first-order solvers (1208.1751).
• Provable convergence rates and a posteriori verifications for SOS-based approaches (1903.04798).
• Integration of data-driven and model-based methods for reducing conservativeness while retaining certifiability (e.g., hybrid approaches using both Lyapunov ellipsoid cores and Bernstein polynomial–modeled cones) (2111.09382).
• Efficient distributed computation of ROA inner approximations in large-scale and networked systems (2009.12345).
• Advanced shape-fitting and scale-invariant objectives for star-convex and semialgebraic sets (2203.13071), improving upon determinant- and trace-based heuristics previously used for ellipsoid fitting.

Challenges remain in extending tractable inner approximation techniques to high-dimensional, highly nonlinear, or hybrid systems, as well as in robustly handling systems with uncertainty, partial observations, or data-driven models.

• D. Henrion, M. Korda, "Convex computation of the region of attraction of polynomial control systems" (1208.1751).
• M. Korda, D. Henrion, C.N. Jones, "Inner approximations of the region of attraction for polynomial dynamical systems" (1210.3184).
• I. Kalur, et al., "Estimating Regions of Attraction for Transitional Flows using Quadratic Constraints" (2103.05426).
• J. Guthrie, "Inner and Outer Approximations of Star-Convex Semialgebraic Sets" (2203.13071).
• B.G. Primbs, "Regions of attraction for nonlinear hybrid systems: A convex programming approach" (related work referenced).
• M. Chesi, "Domain of Attraction: Analysis and Control via SOS Programming".

The literature documents that ellipsoidal inner approximations—particularly as quadratic Lyapunov sublevel sets or as quadratic SOS relaxations—are foundational and widely applicable, but that advanced convex programming and set-theoretic developments continue to extend their reach and accuracy within control and systems theory.